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By C. Kohn
Waterford Agricultural Sciences
How do we know we know?

 A major concern in science is proving that what we have
observed would occur again if we repeated the experiment.
 Random things can affect our experiments.
 Your samples might be affected by little things that change or
skew your results.
 The trends you find in your experiment may not occur in a
different experiment done in the same way.
 We must always be prepared to answer
the Scientist’s Questions:
-How do I know I am not wrong?
-How do I know that this will always occur
every time I do this experiment?
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Target Shooting & Statistics

 Data in research is sort of like target practice.
 When you are shooting at a target, you want all of your
shots to be close together.
 The closer your shots are to each other, the better.
 You also need to take a lot of shots in order to be accurate.
 The more times you shoot at a target, the more accurate of a
shooter you are.
 Statistics are the same: the
more data we have, and the
more similar each number is
to each other, the better.
Source: topendsports.com
Science & Statistics

 In science, we can use statistical equations to determine
whether or not we can be confident in our results.
 In other words, the use of statistics can tell us whether our
experimental results are reliable.
 If we are likely to see similar results every single time, this
means that our results are reliable.
 On the other hand, if we get very different results each time we
do an experiment, our data is less reliable.
 The more variable our data,
the less reliable it is.
Less reliable
 The less our data varies,
the more reliable it is.
More reliable
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The “Real” Average

 When we need to calculate the average of our data (or “mean”), we
can encounter problems with reliability.
 Mean: the numerical average of data (mean = average)
 It is calculated by dividing the sum of the numbers by the sample
size. Mean = (Sum of Data)/(Sample Size)
 E.g. mean of 1,2,&3 would be (1+2+3)/3 = 2. Our mean is 2.
 When we take the average of something, we are using a number
that can change as we gain or lose data.
 For example, imagine if we wanted to
know the mean height of this class.
 To obtain this number, we would…
 1. Record each person’s height
 2. Add them all together, and
 3. Divide by the number of
students we have to get the
“mean height”.
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Averages (cont.)

 However, if we gained or lost a student, the mean (or
average) height would change.
 The “average height” is not one number; it can change!
 If our class did not have very many students, the addition of
one more person’s height would have a big impact on the
calculated average.
 On the other hand, if we had 1000 students in our class, the
addition of one more person’s height would hardly change the
calculated average.
 If the new person’s height was very similar to the average,
our calculated average would not change much.
 On the other hand, if they were 6’7”, our calculated average
would change a lot more.
Factors that Affect Data Reliability

 Things that affect the reliability of our data include:
 How similar our data is:
 The more similar the data, the more reliable our average will
be.
 E.g. if all of our students are between 5’10” and 6’1”, we would
have more reliable data than if the range of the data was greater
(such as if the range was between 4’5” and 7’1”)
 The amount of data we have:
 The more data we have, the more reliable our average will be.
 E.g. if you flip a coin 3 times, you might get 2 heads, 1 tail.
 If you flip a coin 10 times, you might get 6 heads and 4 tails.
 If you flip a coin 100 times, you might get 49 heads, 51 tails
 Each time we get closer to the “real” average of 50/50
goldenstateofmind.com
Examples
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
 For example, let’s imagine you want to how
UV light affects radish growth.
 If you have only six plants, your data will not be very reliable.
 If you have thousands of plants, your data will be much more
reliable.
 If the height of your plants varies a lot (e.g. some are 2 inches,
some are 20 inches), then your data will not be very reliable.
 If all your plants are almost the same size, your data will be very
reliable.
 So how do we know for sure if our data is reliable or not?
Standard Deviation

 Standard Deviation is a measurement of how much our data
varies.
 Low variance means your data is all very similar.
These corn plants would have low SD
 High variance means your data is very dissimilar.
These corn plants would have high SD
 Standard deviation is calculated by the following formula:
SD = √[(dataa-avg)2 + (datab-avg)2…)/(n-1)]
SD = stand. dev
n = sample size
Standard Deviation Example

 For example, let’s pretend that our radish heights were:
6.1 ; 5.8 ; 7.2 ; 4.3 ; 5.5 ; 5.8 cm
 The average (or mean) height would be:
(6.1 + 5.8 + 7.2 + 4.3 + 5.5 + 5.8)/6 = 34.7/6 = 5.8 cm
 To calculate standard deviation (s) we would subtract the mean value
from each individual value, square it, divide by n-1, and take the
square root:
 √[ [(6.1-5.8)2 + (5.8-5.8)2 + (7.2-5.8)2 + (4.3-5.8)2 + (5.5-5.8)2 + (5.8-5.8)2 ]/(6-1)] =
 √[ [ (0.32 ) +
(02 )
 √[ [ (0.09 ) +
(0 )
+
+
(1.42 ) +
(-1.52 ) +
(0.32 ) + (02 ) ] (5) ] =
(1.96) + (2.25 ) + (0.09 ) + (0 ) / 5 ] =
 √[4.39 / 5] = √[0.878] = 0.94 cm
 Our Standard Deviation score is 0.94 cm (note: SD is measured in the same units as our data)
Standard Deviation

 Standard Deviation is a measure of variability.
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 We want our data to be very similar;
we don’t want it to be spread out.
 Data is like butter – we want it in a tight form, like a stick. We
don’t want it to melt and spread everywhere.
 Standard Deviation can be used to find the Margin of Error
(or our “range of accuracy”).
 Margin of Error is usually equal to 2x the Standard Deviation on
either side of the mean (average).
 As you will soon see, we have a better way to calculate the
margin of error.
 The margin of error shows us all of the possible results for our
research (if we repeated it) with a 95% accuracy.
Standard Error

 Standard Deviation is a measure of how varied your data is.
 However, as we said before, both variance and the size of your
sample affect the reliability of your data.
 Standard Deviation is only a measure of variance.
 Standard Error is a measurement of reliability of a data
sample; it involves both the size of your data sample and the
variance of your data.
 Standard Error is calculated by dividing your Standard Deviation
by the square root of your sample size.
 Standard Error = [ SD / √(n) ]
n = your sample size
 Standard Error is a measure of the reliability of your data.
 It uses both the size of the data sample and the variability of the data.
Radish Standard Error Example

 For example, for our hypothetical radishes:
Our 6 radish heights were:
6.1 ; 5.8 ; 7.2 ; 4.3 ; 5.5 ; 5.8 cm
Our mean was 5.8 cm.
Our Standard Deviation was 0.94 cm.
Our Standard Error is 0.94/ √(6) = 0.38 cm
Standard Error and Confidence

 Standard Error is a better way to calculate your Margin
of Error.
 The benefit of using Standard Error for your Margin of
Error is that SE includes the population size as well as
variance
 Again, the lower the variance and the higher the population
size, the more reliable the data.
 Standard Deviation only includes variance
 It does not include population size.
Key Benefits of Standard Error

 Standard Error tells us the likelihood of getting the
same result if your repeated the experiment again.
 For example, if you are very likely to get the same average
if you did the experiment again, you would have a small
Standard Error.
 If your results were more likely to be different, you would
have a large Standard Error.
 We always want to have as small of a Standard Error value
as possible.
Standard Error and Research

 Standard Error is used to give us Error Bars.
 Error bars are a visual depiction of your Margin of Error.
 If the error bars overlap, there is no statistical difference
between the two groups (we have to treat them as if they
are the same).
 E.g. these two groups are statistically the same because the
error bars overlap with each other.
 Even though the blue bar is
bigger, we have to treat them
as if they are the same
because statistically, they are.
Error Bars

 In this example, the control has an average height (or
mean height) that is over a full centimeter taller than the
experimental average.
 It looks as if the blue average is noticeably greater than the
red average.
 However, the Error Bars (+/- 2 Standard Errors) overlap.
 If your error bars overlap, this means that there is no
statistically significant
difference between
the control and the
experimental average.
 You must treat them as
if they are the same.
Error Bars overlap;
they are statistically
the same.
Error Bars do not
overlap; they are
statistically
different.
No matter how many
times we repeated this
experiment, the orange
would always be
shorter on average
than the green.

In this case, if we
repeated the
experiment again,
the red could be
taller on average
than the blue.
Standard Deviation in Excel

 Standard Deviation in Excel:
 Use this formula
=(STDEV(data set cell range)/(your sample size^(1/2)))
 Manipulate your data as needed (e.g. for Standard Error,
divide your Standard Deviation by the square root of your
sample size; multiply this by 2 to get your margin of error).
 For sample sizes larger than 30, a reliable average can be found
within the range of +/- 2 Standard Errors.
 Smaller sample sizes require more complicated calculations
Summary

 The more consistent the data, and the larger the sample size, the more reliable
that data is.
 Vice versa, small populations and highly variable data mean that it is less reliable.
 Mean is the average of the data (Sum of the Data / Sample Size)
 Standard Deviation is a measure of variability
 Margin of Error is the range in which we can be 95% sure of accuracy.
 Standard Error is a of measure the reliability of our data; it includes both
variation and the sample size.
 Error bars can be made on graphs using +/- 2x the Standard Error value.
 Error bars indicate the range of accuracy of that data.
 If the error bars of two graphs overlap, those two graphs are considered
statistically the same.
 The error bars do not overlap, they are statistically different.
Calculating SD and SE

 Step 1: Calculate the average of each group
 Step 2: Subtract the average from each number
 Step 3: Square the result from Step 2 for each number
 Step 4: Add up each squared result
 Step 5: Divide the sum of each squared result by (n-1)
 n = the number of numbers you have
 Step 6: Take the square root of your result from Step 5
 This is your standard deviation
 Step 7: To find standard error, divide your calculated
standard error by the square root of n
Review Concepts

 Definition of: 1) mean, 2) Standard Deviation, 3) Standard
Error, 4) Error Bars
 Relationships between variability and accuracy
 2 factors that increase the reliability of data
 How to tell if two graphs are statistically different
 How to calculate standard error and standard deviation.